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Negative refraction and Left-handed behavior in Photonic Crystals: FDTD and Transfer matrix method studies. Peter Markos, S. Foteinopoulou and C. M. Soukoulis. Outline of Talk. What are metamaterials? Historical review Left-handed Materials Results of the transfer matrix method

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slide1

Negative refraction and Left-handed behavior in Photonic Crystals:

FDTD andTransfer matrix method studies

Peter Markos, S. Foteinopoulou and C. M. Soukoulis

slide2

Outline of Talk

  • What are metamaterials?
  • Historical review Left-handed Materials
  • Results of the transfer matrix method
  • Determination of the effective refractive index
  • Negative n and FDTD results in PBGs (ENE & SF)
  • New left-handed structures
  • Experiments on negative refractions (Bilkent)
  • Applications/Closing Remarks

E. N. Economou & S. Foteinopoulou

slide3

What is an ElectromagneticMetamaterial?

  • A composite or structured material that exhibits properties not found in naturally occurring materials or compounds.
  • Left-handed materials have electromagnetic properties that are distinct from any known material, and hence are examples of metamaterials.
slide4

Electromagnetic Metamaterials

Example: Metamaterials based on repeated cells…

slide5

e,m space

(-,+)

(+,+)

(-,-)

(+,-)

Veselago

We are interested in how waves propagate through various media, so we consider solutions to the wave equation.

Sov. Phys. Usp. 10, 509 (1968)

left handed waves
Left-Handed Waves
  • If then is a right set of vectors:
  • If then is a left set of vectors:
slide7

Energy flux in plane waves

  • Energy flux (Pointing vector):
    • Conventional (right-handed) medium
    • Left-handed medium
frequency dispersion of lh medium
Frequency dispersion of LH medium
  • Energy density in the dispersive medium
  • Energy density Wmust be positive and thisrequires
  • LH medium is always dispersive
  • According to the Kramers-Kronig relations –

it isalways dissipative

slide9

PIM

RHM

NIM

LHM

PIM

RHM

PIM

RHM

2

2

1

1

(1)

(2)

(1)

(2)

“Reversal” of Snell’s Law

slide10

RH

RH

LH

RH

RH

RH

n=1

n=1.3

n=1

n=1

n=-1

n=1

Focusing in a Left-Handed Medium

slide11

PBGs as Negative Index Materials (NIM)

  • Veselago : Materials (if any) withe< 0 and m< 0
  • ·em> 0  Propagation
  • k, E, H Left Handed (LHM)  S=c(E x H)/4p

opposite to k

  • ·   Snell’s law with < 0 (NIM)
  • ·g opposite to k
  • ·Flat lenses
  • ·Super lenses
slide12

S1

S2

O΄Μ

A

B

Ο

Objections to the left-handed ideas

Parallel momentum is not conserved

Causality is violated

Fermat’s Principle ndl minimum (?)

Superlensing is not possible

slide13

Reply to the objections

  • Photonic crystals have practically zero absorption
  • Momentum conservation is not violated
  • Fermat’s principle is OK
  • Causality is not violated
  • Superlensing possible but limited to a cutoff kc or 1/L
slide14

Materials with e< 0 and m <0 Photonic Crystals

opposite to

opposite to

opposite to

opposite to

slide15

Super lenses

is imaginary

  • Wave components with decay, i.e. are lost , then Dmax l

If n < 0, phase changes sign

if

imaginary

thus

ARE NOT LOST !!!

slide17

First Left-Handed Test Structure

UCSD, PRL 84, 4184 (2000)

slide18

Wires alone

Split rings alone

e<0

m<0

m<0

m>0

m>0

m>0

m>0

e<0

e<0

e<0

Wires alone

Transmission Measurements

Transmitted Power (dBm)

6.0

6.5

5.5

7.0

5.0

4.5

Frequency (GHz)

UCSD, PRL 84, 4184 (2000)

slide19

A 2-D Isotropic Structure

UCSD, APL 78, 489 (2001)

slide20

Measurement of Refractive Index

UCSD, Science 292, 77 2001

slide21

Measurement of Refractive Index

UCSD, Science 292, 77 2001

slide22

Measurement of Refractive Index

UCSD, Science 292, 77 2001

slide23

Transfer matrix is able to find:

  • Transmission (p--->p, p--->s,…)p polarization
  • Reflection (p--->p, p--->s,…)s polarization
  • Both amplitude and phase
  • Absorption

Some technical details:

  • Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24
  • Length of the sample: up to 300 unit cells
  • Periodic boundaries in the transverse direction
  • Can treat 2d and 3d systems
  • Can treat oblique angles
  • Weak point: Technique requires uniform discretization
slide24

Structure of the unit cell

EM wave propagates in the z -direction

Periodic boundary conditions

are used in transverse directions

Polarization: p wave: E parallel to y

s wave: E parallel to x

For the p wave, the resonance frequency

interval exists, where with Re meff <0, Re eeff<0 and Re np <0.

For the s wave, the refraction index ns = 1.

Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm

Typical permittivity of the metallic components: emetal = (-3+5.88 i) x 105

slide25

Structure of the unit cell:

SRR

EM waves propagate in the z-direction.

Periodic boundary conditions are used in the xy-plane

LHM

slide26

Left-handed material: array of SRRs and wires

Resonance frequency as a function of metallic permittivity

 complex em

 Real em

slide27

Dependence of LHM peak on metallic permittivity

The length of the system is 10 unit cells

slide30

Example of Utility of Metamaterial

The transmission coefficient is an example of a quantity that can be determined simply and analytically, if the bulk material parameters are known.

UCSD and ISU, PRB, 65, 195103 (2002)

slide31

Effective permittivity e(w) and permeability m(w) of wires and SRRs

UCSD and ISU, PRB, 65, 195103 (2002)

slide33

Effective refractive index n(w) of LHM

UCSD and ISU, PRB, 65, 195103 (2002)

slide34

Determination of effective parameters from transmission studies

From transmission and reflection data, the index of refraction n was calculated.

Frequency interval with Re n<0 and very small Im n was found.

slide36

Another 1D left-handed structure:

Both SRR and wires are located on the same side of the dielectric board.

Transmission depends on the orientation of SRR.

Bilkent & ISU APL 2002

slide37

0.33 mm

w

t»w

t=0.5 or 1 mm

w=0.01 mm

t

0.33 mm

3 mm

l=9 cm

0.33 mm

3 mm

ax

Periodicity:

ax=5 or 6.5 mm

ay=3.63 mm

az=5 mm

Number of SRR

Nx=20

Ny=25

Nz=25

Polarization: TM

y

E

x

y

x

B

z

slide38

New designs for left-handed materials

eb=4.4

Bilkent and ISU, APL 81, 120 (2002)

slide39

ax=6.5 mm

t= 0.5 mm

Bilkent & ISU APL 2002

slide40

ax=6.5 mm

t= 1 mm

Bilkent & ISU APL 2002

slide42

Phase and group refractive index

  • In both the LHM and PC literature there is still a lot of confusion regarding the phase refractive index np and the group refractive index ng. How these properties relate to “negative refraction” and LH behavior has not yet been fully examined.
  • There is controversy over the “negative refraction” phenomenon. There has been debate over the allowed signs (+ /-) for np and ng in the LH system.
slide43

DEFINING phase and group refractive index np and ng

  • In any general case:
  • The equifrequency surfaces (EFS) (i.e. contours of constantfrequency in 2D k-space) in air and in the PC are needed to find the refracted wavevector kf (see figure).
  • vphase=c/|np| and vgroup= c/|ng|
  • Where c is the velocity of light
  • So from k// momentum conservation:|np|=c kf () /.

Remarks

  • In the PC system vgroup=venergy so |ng|>1. Indeed this holds !
  • np <1 in many cases, i.e. the phase velocity is larger than c in many cases.
  • npcan be used in Snell’s formula to determine the angle of the propagating wavevector. In general this angle is not the propagation angle of the signal. This angle is the propagation angle of the signal only when dispersion is linear (normal), i.e. the EFS in the PC is circular (i.e. kf independent of theta).
  • ng can never be used in a Snell-like formulato determine the signals propagation angle.
slide44

Index of refraction of photonic crystals

  • The wavelength is comparable with the period of the photonic crystal
  • An effective medium approximation is not valid

Equifrequency surfaces

Effective index

Refraction angle

kx

ky

w

Incident angle

slide46

Photonic Crystals with negative refraction.

S. Foteinopoulou, E. N. Economou and C. M. Soukoulis

slide49

Schematics for Refraction at the PC interface

EFS plot of frequency a/l = 0.58

slide50

Schematics for Refraction at the PC interface

EFS plot of frequency a/l = 0.535

slide53

Superlensing in 2D

Photonic Crystals

Lattice constant=4.794 mm

Dielectric constant=9.73

r/a=0.34, square lattice

Experiment by Ozbay’s group

slide55

Band structure, negative refraction and experimental set up

Frequency=13.7 GHz

Negative refraction is achievable

in this frequency range for

certain angles of incidence.

Bilkent & ISU

slide59

Subwavelength Resolution in PC based Superlens

The separation between the two point sources is l/3

slide60

Photonic Crystals with negative refraction.

Photonic

Crystal

vacuum

FDTD simulations were used to study the time evolution of an EM wave as it hits the interface vacuum/photonic crystal.

Photonic crystal consists of an hexagonal lattice of dielectric rods

with e=12.96. The radius of rods is r=0.35a. a is the lattice constant.

slide65

Photonic Crystals: negative refraction

The EM wave is trapped temporarily at the interface and after a long time,

the wave front moves eventually in the negative direction.

Negative refraction was observed for wavelength of the EM wave

l= 1.64 – 1.75 a (a is the lattice constant of PC)

slide66

Conclusions

  • Simulated various structures of SRRs & LHMs
  • Calculated transmission, reflection and absorption
  • Calculated meff and eeffand refraction index(with UCSD)
  • Suggested new designs for left-handed materials
  • Found negative refraction in photonic crystals
  • A transient time is needed for the wave to move along the - direction
  • Causality and speed of light is not violated.
  • Existence of negative refraction does not guarantee the existence of
  • negative n and so LH behavior
  • Experimental demonstration of negative refraction and superlensing
  • Image of two points sources can be resolved by a distance of l/3!!!

$$$ DOE, DARPA, NSF, NATO, EU

slide67

Publications:

  • P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002)
  • P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002)
  • D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002)
  • M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, Appl. Phys. Lett. (2002)
  • P. Markos, I. Rousochatzakis and C. M. Soukoulis, Phys. Rev. E 66, 045601 (R) (2002)
  • S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL, accepted (2003)
  • S. Foteinopoulou and C. M. Soukoulis, submitted Phys. Rev. B (2002)
  • P. Markos and C. M. Soukoulis, submitted to Opt. Lett.
  • E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, submitted to Nature
  • P. Markos and C. M. Soukoulis, submitted to Optics Express
the keen interest to the topic
The keen interest to the topic
  • Left-Handed Medium (LH)
  • Metamaterial
  • Backward Medium (BW)
  • Double Negative Medium (DNG)
  • Negative Phase Velocity (NPV)
  • Materials with Negative Refraction (MNR)
  • Terminology
slide74

Menu

LHM

Cal

Display

Store

Network Analyzer

Experimental Setup

slide78

Dependence on the incident angle

Transmission peak does not depend

on the angle of incidence !

Transition peak strongly depends

on the angle of incidence.

This structure has an additional

xz - plane of symmetry

slide79

Transmission depends on the orientation of SRR

Transmission properties depend on the orientation of the SRR:

  • Lower transmission
  • Narrower resonance interval
  • Lower resonance frequency
  • Higher transmission
  • Broader resonance interval
  • Higher resonance frequency
slide80

Dependence of the LHM T peak on the Im eBoard

In our simulations, we have:

Periodic boundary condition,

therefore no losses due to

scattering into another direction.

Very high Im emetal therefore

very small losses in the metallic

components.

Losses in the dielectric board are crucial for the transmission

properties of the LH structures.

slide82

Superprism Phenomena in Photonic Crystals Experiment

  • H.Kosaka, T.Kawashima et. al. Superprism phenomena in photonic crystals, Phys. Rev. B 58, 10096 (1998)
slide83

Scattering of the photonic crystalHexagonal 2D photonic crystal

    • M.Natomi, Phys. Rev. B 62, 10696 (2000)
  • Using an equifrequency surface (EFS) plots

Vanishingly small index modulation

Small index modulation

slide84

Photonic crystal as a perfect lens

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry

Phys. Rev. B, 65, 201104 (2002)

Resolution limit 0.67 l